Find the domain of function `f(x)=(log)_4[(log)_5{(log)_3(18 x-x^2-77}]`

Options

A. (8,10)
B. [8,10]
C. `(-oo,8]`
D. `[10,oo)`

Answer

A

Explanation

Since log x is defined for ` x gt 0`. Therefore , `f(x) =log_(4)[log_(5){log_(3)(18x-^(2)-77)}]` exists , if ` log_(5){log_(3)(18x-x^(2)-77)} gt 0`
`implies log_(3)(18x-x^(2)-77)} gt 5^(0)`
`implies (18x-x^(2)-77) gt 3^(1)`
`implies x^(2)-18+80 lt 0`
`implies (x-8)(x-10) lt 0 implies 8 lt x lt 10 implies x in (8,10)`
Hence, the domain of definition of the given function is (8,10).